Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

Lijie Li

doi:10.1155/2014/480241

Abstract

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

Highlights

The almost periodic type solutions to stochastic differential equations are among the most attractive topics in mathematical analysis due to their extensive applications in areas such as physics, economics, mathematical biology, and engineering
This is more complicated than the previous case because of the involvement of the Brownian motion W
L is continuous and maps AA(R+; L2(P, H)) into itself, where L is defined by Lemma 12

Summary

Introduction

The almost periodic type solutions to stochastic differential equations are among the most attractive topics in mathematical analysis due to their extensive applications in areas such as physics, economics, mathematical biology, and engineering. This paper is mainly focused on the existence and uniqueness of square-mean almost automorphic mild solutions to the following stochastic functional integrodifferential equations in the abstract form: dN (t, x (t)) = AN (t, x (t)) dt + ∫ B (t − s) N (s, x (s)) ds dt. In [15], Chang et al established a new composition theorem for square-mean almost automorphic functions under conditions which were different from Lipschitz conditions in the literature They apply this new composition theorem to investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation. By virtue of new composition theorem in [15] together with Schauder’s fixed point theorem, we investigate the existence of square-mean almost automorphic mild solutions for a stochastic differential equation in a real separable Hilbert space, which is different from Lipschitz condition in the literature. We discuss the existence and uniqueness of an almost automorphic mild solution to a concrete integrodifferential equation, which is an illustration to demonstrate our main analyses

Preliminaries

Some Lemmas

Existence of Square-Mean Almost Automorphic Solutions

Example